Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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![Three cylindrical rollers of equal length and material are arranged as shown in end view
below. Cylinders 2A and 2B have radiusr and cylinder 1 has radius 4r. Their axes are parallel
to each other and to that of the fixed concave cylindrical surface, of radius 6r, that supports
them. They roll without slip. Assuming no damping and following the steps below,
determine the frequency of small free rolling oscillations.
(a)
For rotation 6, of the cylinder 1, determine the resulting rotation of the lower
cylinders. [Hint: Does it look like an epicyclic gear train?]
When the upper roller rotates with velocity 6, determine the translational and
rotational velocities of the centre of mass of the lower rollers. [Hint: In an epicyclic
gear train the centres of these would be supported on the planet carrier.]
(b)
(c)
Write an expression for the kinetic energy of the whole system at
some instant during the oscillation when the rotational velocity of cylinder 1 is , .
(d)
Write an expression for the gravitational potential energy of the system when the
cylinder 1 has rotated through angle 0, from its equilibrium position. Use the
equilibrium position as the potential energy datum.
(e)
Assuming no damping, apply the principle of conservation of energy to the system
to derive the governing differential equation of motion. [Energy]=0
(f)
By assuming small oscillation amplitude, linearise the differential equation
obtained in part (e). Then determine the natural frequency of oscillation of the
oscillating motion.](https://content.bartleby.com/qna-images/question/18f521d1-acf6-411c-a6b0-35837d93d501/a66bf1ca-eec2-423c-a8f7-d47dd5e09bf2/kovzg6n_processed.png)
Transcribed Image Text:Three cylindrical rollers of equal length and material are arranged as shown in end view
below. Cylinders 2A and 2B have radiusr and cylinder 1 has radius 4r. Their axes are parallel
to each other and to that of the fixed concave cylindrical surface, of radius 6r, that supports
them. They roll without slip. Assuming no damping and following the steps below,
determine the frequency of small free rolling oscillations.
(a)
For rotation 6, of the cylinder 1, determine the resulting rotation of the lower
cylinders. [Hint: Does it look like an epicyclic gear train?]
When the upper roller rotates with velocity 6, determine the translational and
rotational velocities of the centre of mass of the lower rollers. [Hint: In an epicyclic
gear train the centres of these would be supported on the planet carrier.]
(b)
(c)
Write an expression for the kinetic energy of the whole system at
some instant during the oscillation when the rotational velocity of cylinder 1 is , .
(d)
Write an expression for the gravitational potential energy of the system when the
cylinder 1 has rotated through angle 0, from its equilibrium position. Use the
equilibrium position as the potential energy datum.
(e)
Assuming no damping, apply the principle of conservation of energy to the system
to derive the governing differential equation of motion. [Energy]=0
(f)
By assuming small oscillation amplitude, linearise the differential equation
obtained in part (e). Then determine the natural frequency of oscillation of the
oscillating motion.
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- A 11-lb wheel is mounted on a 2-lb plate whose center is attached to a 1.6 in diameter, 30 in long (G = 11606.22 kips/in^2) steel bar, which is fixed at its other end_ The centroidal polar moment of inertia of the plate is 220 slugs-in^2. The period of torsional oscillation of this assembly is 0.15 s. What is the polar moment of inertia of the wheel?arrow_forwardA thin, circular ring of mass m and radius r is suspended at 0. Determine the natural frequency of small oscillations of this ring in the xy plan. y Hint: The moment of inertia of a thin, circular ring of mass m and radius r about its central axis is I = mr². In this problem, the ring rotates about point 0; therefore, we need to use the Parallel Axis Theorem.arrow_forwardWhich one is correct please?arrow_forward
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